Exponential Sums of the Titan Polynomial: Numerical Evidence for Non-Generic Monodromy
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866902310662176768 |
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| author | Chen, Ruqing |
| author_facet | Chen, Ruqing |
| contents | <p>We compute the additive exponential sums S_p = Σ exp(2πi Q(n)/p) for the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 cyclotomic norm form satisfying the palindromic symmetry Q(n) = Q(1−n).</p> <p>For the 111 primes p ≡ 1 (mod 47) up to 50,000, the normalized magnitude |S_p|/√p has observed mean ≈ 1.50 and maximum ≈ 8.65 (at p = 283). This mean lies far below both the Gaussian random walk prediction (≈ 5.97 for 45 independent unit vectors) and the generic compact symplectic group USp(44) prediction (≈ 3.74). The palindromic symmetry selects symplectic over orthogonal monodromy; by Katz's theorem, USp(44) is the expected generic group. The dramatic shortfall suggests the geometric monodromy group is a proper subgroup of USp(44), constrained by the cyclotomic origin Q(n) = Φ₄₇(n/(n−1)).</p> <p>We conjecture that this anomalous cancellation arises from a Jacobian splitting: the associated algebraic variety decomposes into lower-dimensional CM abelian sub-varieties indexed by the characters of (ℤ/47ℤ)×. The repository includes the paper (5 pages), two data CSVs (111 data points and summary statistics), a 4-panel PDF figure, and three scripts (Python computation, Python figure generation, SageMath original).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18526918 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Exponential Sums of the Titan Polynomial: Numerical Evidence for Non-Generic Monodromy Chen, Ruqing exponential sums Frobenius eigenvalues monodromy group Katz-Sarnak philosophy Weil bound cyclotomic norm form Sato-Tate distribution symplectic group Jacobian splitting complex multiplication palindromic polynomial Titan polynomial Mathematics Number Theory <p>We compute the additive exponential sums S_p = Σ exp(2πi Q(n)/p) for the Titan polynomial Q(n) = n⁴⁷ − (n−1)⁴⁷, a degree-46 cyclotomic norm form satisfying the palindromic symmetry Q(n) = Q(1−n).</p> <p>For the 111 primes p ≡ 1 (mod 47) up to 50,000, the normalized magnitude |S_p|/√p has observed mean ≈ 1.50 and maximum ≈ 8.65 (at p = 283). This mean lies far below both the Gaussian random walk prediction (≈ 5.97 for 45 independent unit vectors) and the generic compact symplectic group USp(44) prediction (≈ 3.74). The palindromic symmetry selects symplectic over orthogonal monodromy; by Katz's theorem, USp(44) is the expected generic group. The dramatic shortfall suggests the geometric monodromy group is a proper subgroup of USp(44), constrained by the cyclotomic origin Q(n) = Φ₄₇(n/(n−1)).</p> <p>We conjecture that this anomalous cancellation arises from a Jacobian splitting: the associated algebraic variety decomposes into lower-dimensional CM abelian sub-varieties indexed by the characters of (ℤ/47ℤ)×. The repository includes the paper (5 pages), two data CSVs (111 data points and summary statistics), a 4-panel PDF figure, and three scripts (Python computation, Python figure generation, SageMath original).</p> |
| title | Exponential Sums of the Titan Polynomial: Numerical Evidence for Non-Generic Monodromy |
| topic | exponential sums Frobenius eigenvalues monodromy group Katz-Sarnak philosophy Weil bound cyclotomic norm form Sato-Tate distribution symplectic group Jacobian splitting complex multiplication palindromic polynomial Titan polynomial Mathematics Number Theory |
| url | https://doi.org/10.5281/zenodo.18526918 |